Problem 32 - Advanced Engineering Mathematics

Question

In Problems \(25-32\), verify the given identity. Assume continuity of all partial derivatives. $$ \operatorname{curl}(\operatorname{curl} \mathbf{F}+\operatorname{grad} f)=\operatorname{curl}(\operatorname{curl} \mathbf{F}) $$

Step-by-Step Solution

Answer

The identity is verified; both sides are equal to \(\operatorname{curl}(\operatorname{curl} \mathbf{F})\).

Step 1: Understand the Identity

The given identity is \(\operatorname{curl}(\operatorname{curl} \mathbf{F} + \operatorname{grad} f) = \operatorname{curl}(\operatorname{curl} \mathbf{F})\). This needs to be verified under the assumption that all partial derivatives are continuous.

Step 2: Recall Vector Calculus Identities

Two important identities are helpful here: 1) \(\operatorname{curl}(\operatorname{grad} f) = 0\) and 2) \(\operatorname{curl}(\mathbf{A} + \mathbf{B}) = \operatorname{curl}\mathbf{A} + \operatorname{curl}\mathbf{B}\). These will be used to simplify the left-hand side of the identity.

Step 3: Apply Curl to Grad

Apply the identity \(\operatorname{curl}(\operatorname{grad} f) = 0\). Therefore, \(\operatorname{curl}\operatorname{grad} f = 0\) simplifies the term in the equation.

Step 4: Simplify the Left-Hand Side

Using the linearity of the curl operator, simplify the left-hand side: \(\operatorname{curl}(\operatorname{curl} \mathbf{F} + \operatorname{grad} f) = \operatorname{curl}(\operatorname{curl} \mathbf{F}) + \operatorname{curl}(\operatorname{grad} f)\). Since \(\operatorname{curl}(\operatorname{grad} f) = 0\), this reduces to \(\operatorname{curl}(\operatorname{curl} \mathbf{F})\).

Step 5: Conclude the Verification

After simplification, both sides of the identity become \(\operatorname{curl}(\operatorname{curl} \mathbf{F})\). Thus, the given identity is verified.

Key Concepts

Curl of a Vector Field

The curl of a vector field is a fundamental concept in vector calculus, often used to measure the rotation of a field. If you imagine a vector field as a flow of water, the curl represents the swirling or rotational aspect of that flow. Mathematically, for a vector field \( \mathbf{F} = (F_1, F_2, F_3) \), the curl is given by the expression:

• \( \operatorname{curl} \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \)

This operation results in another vector field which highlights the tendency of particles in the field to rotate around a point.
Understanding the physical interpretation of curl helps in visualizing fields encountered in electromagnetism and fluid dynamics.

Gradient

The gradient is an operator used in vector calculus to express the rate and direction of change in a scalar field. It transforms a scalar function into a vector field. For a scalar function \( f(x, y, z) \), the gradient \( \operatorname{grad} f \) is given by:

• \( \operatorname{grad} f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \)

The result is a vector that points in the direction where the function increases most rapidly. Its magnitude is the rate of increase in that direction.
Using the gradient transforms the problem of finding maximum increase in a scalar field into understanding vector fields, which is intuitive for visualizing optimization problems and understanding physical phenomena like heat flux or electric potential.

Vector Identities

Vector identities are essential shortcuts in vector calculus that can greatly simplify computations. They combine operations like divergence, gradient, and curl in predictable ways. A couple of crucial vector identities used in solving exercises like the one given include:

• \( \operatorname{curl}(\operatorname{grad} f) = 0 \)
• \( \operatorname{curl}(\mathbf{A} + \mathbf{B}) = \operatorname{curl} \mathbf{A} + \operatorname{curl} \mathbf{B} \)

These identities are useful for reducing complex multivariable calculus problems into simpler forms. Understanding why these identities hold involves recognizing properties like the continuity of partial derivatives and the inherent nature of rotational dynamics in vector fields. These identities facilitate efficient problem-solving and deepen understanding of field behavior in physical systems.

Partial Derivatives

Partial derivatives are a core component of multivariable calculus, allowing for the analysis of functions with multiple variables. When a function depends on several variables, a partial derivative describes how the function changes as only one of those variables changes, keeping others constant.
For a function \( f(x, y, z) \), the notation for partial derivatives is:

• \( \frac{\partial f}{\partial x} \), \( \frac{\partial f}{\partial y} \), \( \frac{\partial f}{\partial z} \)

These derivatives supply a way to assess and describe local behavior in multidimensional spaces, capturing directional change. They are critical in laying the groundwork for further operations like curl, divergence, and gradient.
Understanding partial derivatives and their interaction is crucial to solving problems involving change, optimization, and the behavior of vector fields across continuous domains.